Image Texture is duplicating itself across the back of the image?
In the unwrapped sphere, the center of the flower pattern is probably where you want the iris to show up.  The "pedals" represent the sides and back of the sphere.  Blender automatically wraps the "pedals" around over the image again if they go off the edge.  On the other hand, if they don't go off the edge, the iris is too large.  Probably the easiest solution is to edit the iris image itself to make the iris smaller and then have more white background.

The original iris picture I used (https://image.shutterstock.com/image-illustration/illustration-blue-iris-digital-artwork-260nw-566777233.jpg) had the white just barely fitting around the iris.  I used Gimp to shrink the iris and make more white margin.

When I used the new iris picture for the texture, it worked.



As you can see, the iris is still huge, but that would have been solved if I had made the iris smaller in Gimp.  You can probably use most other image editors to shrink the iris (maybe even blender, though I couldn't figure out any easy way to do that).  If you have Gimp, I made the iris smaller by selecting the iris and shrinking it with the scale tool.  Then I selected the main (Iris.png) layer and filled the alpha area with white.